The class blog for Math 3010, fall 2014, at the University of Utah

Tag Archives: history

Almost everyone has heard of Florence Nightingale: the Nurse; very few people have heard about Florence Nightingale: the Statistician. Ironically, however, she is one of the most important statisticians to have ever graced the field. Yes, her improvements to sanitation were revolutionary, and surely saved countless lives, but how she was able to bring about those improvements was equally innovative.

Picture this: It’s 1855 during the Crimean War. The air is rank and humid, filled with the smell of blood and sulfur—the fresh, salty aroma of the adjacent Black Sea long forgotten. Although you are required to help care for the many wounded soldiers, you are also charged with collecting, accurately keeping, and even analyzing army mortality records.

After months of attending to these records and conducting your analyses, you are faced with an appalling, yet undeniable, truth: more soldiers are dying from poor sanitation than from combat. In fact, the mortality rates from disease are so great that “during the first seven months of the Crimean campaign, a mortality rate of 60 per cent. per annum from disease alone occurred, a rate of mortality which exceeded even the Great Plague in London…”[1]

Florence Nightingale (FN) believed that allowing 16,000 men[2] to die from causes that were easily prevented with improved sanitation was almost akin to murder, and it would be equally criminal to do nothing to prevent these needless deaths from happening again[3]. She also felt it was downright disgraceful, if not scandalous, for a nation that considered itself the epitome of civilization to be this neglectful of their sanitation policy[4]. Sadly, these poor sanitary conditions were not just associated with the battlefield. As the war ended and FN returned home, she found that army barracks and even the hospitals experienced equally appalling mortality rates from disease[5].

She knew sanitation policies needed to be improved. She knew her statistical analysis was the best tool she could use to convince others of this need. Yet, she also knew that she would have to develop a better way to convey her data to the general public. She realized that although the empirical evidence would easily convince those who knew how to read the data, publishing it in the traditional way would severely limit the amount of people who could actually utilize the information. A smaller group of supporters meant that it would take more time to bring about the necessary improvements, and more time meant that more people would die. No, if her campaign was to succeed, she needed to have something that enabled almost everyone to easily draw the same conclusion she had for herself: their nation’s sanitation methods were claiming more lives than their enemy’s artillery.

She decided to model her data as a graphical illustration, specifically as a Rose chart (also known as a Polar Area or Coxcomb chart).

Fig. 2Nightingale’s 1858 Rose Chart that graphically illustrates mortality rates Text:The Area of the blue, red, & black wedges are each measured from the centre as the common vertex. The blue wedges measured from the centre of the circle represent area for area the deaths from Preventible or Mitigable Zymotic diseases; the red wedges measured from the centre the deaths from wounds; & the black wedges measured from the centre the deaths from all the other causes. The black line across the red triangle in Nov. 1854 marks the boundary of the deaths from all other causes during the month. In October 1854, & April 1855; the black area coincides with the red; in January & February 1856, the blue coincides with the black. The entire areas may be compared by following the blue, the red & the black lines enclosing them.

At the time, these illustrations were quite an extraordinary feat, and very few statisticians had previously attempted to use these representations. This is because if the statistician was not careful with their calculations, then the representations could very easily mislead the observer. In fact, even FN was mislead at one point. Her initial analysis of the army’s mortality records led her to conclude that it was malnutrition, and not sanitation, that was the major cause of death during the Crimean campaign[6].

As can be seen inthis link here[7], a Rose chart consists of multiple wedges. These wedges are called sectors and each one represents a different category in one’s data. It is important to note that the individual value of each category is represented by the sector’s area and not the radius. This was actually what caused FN to misinterpreted her data at first, as she used the radius, instead of the area, to represent the value of each sector. Despite the fact that the calculations can make it tricky to accurately represent numerical data, the chart can, on-the-other-hand, simplify comparisons and enable observers to easily identify causation[8].

Florence Nightingale was a true devotee to both statistical analysis and improved health care. She was an innovative woman, who pioneered the means with which to effectively communicate statistics’ findings that describe social phenomena. She was both revered and admired by her compatriots in the field of mathematics, and her great admiration for statistics earned her the nickname, “The Passionate Statistician.”

Sources:

[1] Pearson, Egon Sharpe, Maurice George Kendall, and Robert Lewis Plackett, eds. Studies in the History of Statistics and Probability. Vol. 2. London: Griffin, 1970.

[2] Nightingale, F. (1858) Notes on Matters Affecting the Health, Efficiency, and Hospital Administration of the British Army Harrison and Sons, 1858

Mathematics is no stranger to unsolved problems. Time and time again, equations, conjectures, and theorems have stumped mathematicians for generations. Perhaps the most famous of these problems was Fermat’s Last Theorem, which stated there is no solution for the equation xp+yp=zp, where x, y, and z are positive integers and p is an integer greater than 2. Pierre de Fermat proposed this theorem in 1637, and for over three hundred fifty years, it baffled mathematicians around the globe. It was not until 1994 that Andrew Wiles finally solved the centuries-old theorem.

Though the most famous, Fermat’s Last Theorem was by no means the only unsolved problem in mathematics. Many problems remain unsolved to this day, driving many institutions throughout the world to offer up prizes for the first person to present a working solution for any of the problems. Some few are general questions, such as “Are there infinitely many real quadratic number fields with unique factorization?” However, most of the problems are specific equations proposed by a single or multiple mathematicians and are generally named after their proposer(s), such as the Jacobian Conjecture or Hilbert’s Sixteenth Problem.

One such problem, proposed by Henri Poincaré in 1904 and thus named the Poincaré Conjecture, remained unsolved until 2002. In order to encourage work on the conjecture, the Clay Mathematics Institute made it a part of the Millennium Problems, which included several of the most difficult mathematics problems without proofs. A proof to any of the problems, including the Poincaré Conjecture, came with a reward of one million US dollars. To this day, the Poincaré Conjecture remains the only problem solved.

The Poincaré Conjecture is a problem in geometry but concerns a concept that, for many, is difficult to comprehend and all but impossible to visualize. The best means to approach it is to imagine a sphere, perfectly smooth and perfectly proportioned. Now, imagine an infinitesimally-thin, perfectly flat sheet of cardboard cuts into the sphere. If you were to take a pen and draw on the cardboard where the sphere and the cardboard intersect, you would produce a circle. If you were to take the sheet of cardboard and move it up through the sphere, the circle where it and the sphere intersect would gradually shrink. Eventually, just as the cardboard is at the edge of the sphere, the circle will have shrunk to a single point.

Note that in the field of topology, this visualization applies to any shape that is homeomorphic to a three dimensional sphere (referred to as a 2-sphere in topology since its surface locally looks like a two dimensional plane, much as how the Earth appears flat while standing on its surface). Homeomorphic refers to a concept in the field of topology concerning, what is essentially, the distortion of a shape. For instance, one of the simplest examples in three dimensions is that a cube is homeomorphic to a sphere, since if you were to compress and mold the cube (much as you would your childhood PlayDoh), you could eventually shape it into a sphere. However, in topology, you are not allowed to create or close holes in a shape. This is why shapes such as a donut or a cinder-block are not homeomorphic to a sphere, due to the holes that go through them.

Poincaré proposed a concept concerning homeomorphism and the previously described visualization, and it is here where imagining the problem no longer becomes possible. We live in a three-dimensional world, where any position in space can be plotted based on relativity to three axes, all perpendicular to each other. To imagine a fourth spatial dimension perpendicular to those three is mentally impossible, as is any shape with higher dimensions, and yet many problems in geometry and physics relate to a fourth and even higher dimensions. The Poincaré Conjecture relates to these higher dimensional shapes, specifically closed 3-manifolds (shapes with a locally three dimensional surface). It states that, if a loop can be drawn on a closed 3-manifold and then be constricted to a single point, much like the intersection of the cardboard plane and the sphere in the aforementioned example, then the closed 3-manifold is homeomorphic to a 3-sphere, the set of points equidistant from a central point in four dimensions (Morgan).

If the concept of the Poincaré Conjecture is difficult to conceive, its solution by Russian mathematician Grigori Perelman in 2002 is almost incomprehensible. Due to the number of variables involved, one could not simply set up a system of equations between a three-dimensional space and a 3-sphere. Instead, Perelman used a differential geometry concept called Ricci Flow, developed by American mathematician Richard Hamilton. In short, it is a system which automatically contracts to a point on any surface, and it proved to be the precise tool needed to prove the Poincaré Conjecture. (THIS video does a good job of explaining it in layman’s terms) (Numberphile)

An example of Ricci flow. Image: CBM, via Wikimedia Commons.

Interestingly, despite the immense difficulty of solving such an abstract problem as the Poincaré Conjecture, Perelman refused the prize awarded to him for his accomplishment. His solution to the problem was an exercise in his own enjoyment, and as he later stated upon being offered the Fields Medal (the mathematician equivalent of the Nobel) and the immense monetary prize, “I’m not interested in money or fame; I don’t want to be on display like an animal in a zoo.” Later, he also argued that his contribution to the solution of the Poincaré Conjecture was “no greater than that of… Richard Hamilton,” and that he felt the organized mathematical community was “unjust.” (BBC News, Ritter)

To this day, the Poincaré Conjecture remains the only Millennium Problem solved. Its proof wound up leading to the solution of various other related geometrical problems and closed a century-old mystery. As the field of mathematics continues to grow and progress, it is only a matter of time until other unsolved problems come to resolution.

Like most people, I end up having dinner with my family every so often. Unlike most people, our family conversations always seem to include math or physics. Generally we end up trying to stump each other with various questions. It’s like a game to us.

A few nights ago we had one of those dinners. I found a lull in the conversation and I fired off, “Hey dad, do you know any significant female mathematicians?” His jaw went a little loose and he gave me a blank stare. It was the same look people give me when they find out I talk about math at the dinner table.

My crippling question had developed from a classroom discussion about an influential female mathematician, Sophie Germain. I’d realized in that discussion that I could name several influential female scientists. For whatever reason, I’d never heard of any women known for their mathematics.

As the feeling of triumph settled in, a smile developed across my face. Sophie and I were about to taste victory. I watched confidently as my father’s eyes slowly rolled back into his head. Gradually his eyes came back down and his smile met mine. I knew I was in trouble when he said, “Well, the oldest one I can think of is Hypatia”.

He went on to tell us the story of the daughter of Theon of Alexandria. Theon was a known scholar and professor of mathematics at the University of Alexandria. His daughter, Hypatia, was given all the best. In particular, her education was second to none. With such great influences, many historians believe she was able to eclipse her father’s knowledge at an early age. In time, people would come from distant cities to learn from her.

Sadly, none of her original work has survived to this day. As a mathematician we remember her largely for her editing and insightful comments on other great works of the time. Some of the more important works included the Arithmetica by Diophantus and Conics of Apollonius. The book on conics was particularly significant. It contained progressive information about cutting cones with planes that helped develop ellipses, hyperbolas and parabolas.

Unfortunately, Hypatia is possibly better known for the way she died than the way she lived. During her lifetime, the quality of life in Alexandria was on the decline. Fighting had developed among the different religious factions and it threatened to destroy the city. At one point, the Roman emperor ordered the destruction of all pagan temples. As an educated pagan that often spoke about non-Christian philosophy, Hypatia was a likely target. Ultimately her end came when a group of Christians pulled her from her carriage, drug her into a church, stripped her, beat her to death, tore her to pieces, burned the pieces and disgracefully scattered her remains.

At this point my mother had made it to the other side of the table. She began slapping my father’s arm while muttering something through clenched teeth. “Ok, Ok”, he conceded and continued, “My favorite was Emmy Noether anyway”. “She died of natural causes!” he taunted my mother as she wandered into the kitchen. By this time my smile was long gone. He’d already won the game and was just showing off.

Much like Hypatia, Noether was the daughter of a successful professor of mathematics. Her German family was quite wealthy and provided all her needs. Unfortunately, sometimes society isn’t as helpful.

Noether found herself suffering from restricted access to the University of Erlangen. Her father was on faculty there and I expect it felt like home. Fortunately, she was able to audit classes until they started accepting female students. I’m just speculating here, but I’d like to think she helped drive the decision. A brief four years later she was awarded her Ph.D. summa cum laude.

In spite of her obvious achievement, society still wasn’t quite ready for Noether. She spent the next several years working with David Hilbert and other prominent mathematicians at Göttingen University. Due to a lack of insight, much of the faculty refused to allow a female member. As a result, she worked for free.

As the years went by at Göttingen she worked on incredible theories and taught classes in Hilbert’s name. Eventually her situation improved when she was allowed to become a licensed lecturer. The university wasn’t paying her yet, but at least she could collect student fees. Years later she was finally granted a position as an adjunct teacher.

It’s worth mentioning she was considered a remarkable instructor. More than a few of her students went on to make significant contributions to mathematics. If all things were equal, it would be hard to imagine her without tenure.

While Noether may not be well known to the general public, great minds have given her enormous respect. Einstein referred to her as a “significant creative mathematical genius”. This may well be due to her contributions to general relativity.

Her major mathematical contribution was in a 1921. It was a groundbreaking paper on the theory of ideal rings. She was able to think of things in an extremely abstract way. Many consider her the mother of modern algebra.

One of the more easily understandable ideas she developed involved creating ascending or descending chains. Imagine a set A1 is contained in a set A2, which is contained in a set A3, etc. Sometimes you can see that after a certain point, the rest are the same. For example we might notice that A10, A11, A12, etc are all the same. On the surface it may not seem like a big deal. But techniques like these are frequently exploited in proofs.

Quite honestly, the scope of her contribution is breathtaking. It warms my heart to know people have been making such impressive advances in the last hundred years. It’s even more rewarding to see remarkable individuals like Emmy Noether push through an unjust barrier. In spite of never being treated equally, she emerge a champion with the landscape changed behind her.

By the time my father had finished his glowing praise of Emmy Noether, my face was in my palm. I had clearly lost my challenge. I thought we had him, Sophie.